Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data

This paper concerns the large time behavior of strong and classical solutions to the two-dimensional Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the two-dimensional Stokes approximation equations for the compressible flows with large external potential force, together with a Navier-slip boundary condition, for arbitrarily large initial data. Under the conditions that the corresponding steady state exists uniquely with the steady state density away from vacuum, we prove that the density is bounded from above independently of time, consequently, it converges to the steady state density in L^p and the velocity u converges to the steady state velocity in W^1,p for any 1p<∞ as time goes to infinity; furthermore, we show that if the initial density contains vacuum at least at one point, then the derivatives of the density must blow up as time goes to infinity.     

Weak interaction limit for nuclear matter and the time-dependent Hartree-Fock equation

We consider an effective model of nuclear matter including spin and isospin degrees of freedom, described by an N-body Hamiltonian with suitably renormalized two-body and three-body interaction potentials. We show that the corresponding mean-field theory (the time-dependent Hartree-Fock approximation) is “exact” as N tends to infinity.     

Spontaneous particle creation within the external field approximation of QED

It is expected that strong electromagnetic fields in QED may lead to destabilization of the vacuum, its decay and spontaneous production of an electron-positron pair if the strength of the electric field exceeds some threshold. We study the problem within an external field approximation of QED in presence of time-dependent external fields and define the spontaneous particle creation via the adiabatic limit. We consider some model fields, which we solve analytically or numerically, in order to explain the subtlety of the effect and to recognize problems appearing in its proof. We conclude by characterizing when the effect can exist in a stable way and when it becomes unstable w.r.t. small perturbations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotics of Multivariate Randomness Statistics

Kiefer considered the asymptotics of q-sample Cramer-Von Mises statistics for a fixed q and sample sizes tending to infinity. For univariate observations, McDonald proved the asymptotic normality of these statistics when q goes to infinity while the sample sizes stay fixed. Here we define a class of multivariate randomness statistics that generalizes the class considered by McDonald. We also prove the asymptotic normality of such statistics when the sample sizes stay fixed while q tends to infinity.     

Character expansion method for the first order asymptotics of a matrix integral

The estimation of various matrix integrals as the size of the matrices goes to infinity is motivated by theoretical physics, geometry and free probability questions. On a rigorous ground, only integrals of one matrix or of several matrices with simple quadratic interaction (called AB interaction) could be evaluated so far (see e.g. [19], [17] or [9]). In this article, we follow an idea widely developed in the physics literature, which is based on character expansion, to study more complex interaction. In this context, we derive a large deviation principle for the empirical measure of Young tableaux. We then use it to study a matrix model defined in the spirit of the ‘dually weighted graph model’ introduced in [13], but with a cutoff function such that the matrix integral and its character expansion converge. We prove that the free energy of this model converges as the size of the matrices goes to infinity and study the critical points of the limit.     

Un résultat de stabilité pour les solutions faibles des équations des fluides tournants

We study the limit of the periodic, incompressible, rotating fluid equations, as the Coriolis force goes to infinity: in the case of well-prepared initial data in L², the weak solutions converge to the solution of a two-dimensional, incompressible Navier-Stokes equation. We also prove that the rotating fluid equations are globally well-posed under an appropriate assumption on the oscillating part of the initial data.     

Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential

Abstract

We prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.     

Forward speed motions of a submerged body. The cauchy problem

We consider the linearized problem for the ideal fluid flow induced by the horizontal motion of a fully immersed body. The system of equations is made up of an elliptic problem (P) and an initialvalue problem (R) which are coupled by a pseudo-differential operator T. We define a regularized Cauchy problem (R?) using the Yosida approximation of T; we give energy and wave resistance estimates and finally we obtain existence uniqueness and regularity of the weak solution of (R) by taking the limit when ? goes to zero.     

Remark on asymptotic behaviors of solutions to a quasi-linear elliptic Dirichlet problem with large diffusion

We study asymptotic behaviors of nontrivial solutions to the Dirichlet problem of a quasi-linear elliptic equation and obtain a lower bound for growth of L^∞-norm of the solutions, which implies the L^∞-norm of the solutions goes to infinity as the diffusion coefficient goes to infinity.     

Hamilton cycles in 3‐out

Let G_3‐out denote the random graph on vertex set [n] in which each vertex chooses three neighbors uniformly at random. Note that G_3‐out has minimum degree 3 and average degree 6. We prove that the probability that G_3‐out is Hamiltonian goes to 1 as n tends to infinity. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Eigenvalues of Euclidean random matrices

We study the spectral measure of large Euclidean random matrices. The entries of these matrices are determined by the relative position of n random points in a compact set Ω_n of ?^d. Under various assumptions, we establish the almost sure convergence of the limiting spectral measure as the number of points goes to infinity. The moments of the limiting distribution are computed, and we prove that the limit of this limiting distribution as the density of points goes to infinity has a nice expression. We apply our results to the adjacency matrix of the geometric graph. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Periodic Operators in High-Contrast Media and the Integrated Density of States Function

ABSTRACT

This paper studies the asymptotic behavior for the integrated density of states function for operators associated with the propagation of classical waves in a high-contrast, periodic, two-component medium. Consider a domain Ω₊ contained in the hypercube [0, 2π) ⁿ . We define a function χ_τ which takes the value 1 in Ω₊ and the value τ in [0, 2π) ⁿ \ Ω₊. We extend this setup periodically to ? ⁿ and define the operator L _τ = ??χ_τ ?. As τ goes to infinity, it is known that the spectrum of L _τ exhibits a band-gap structure and that the spectral density accumulates at the upper endpoints of the bands. We establish the existence and some important properties of a rescaled integrated density of states function in the large coupling limit which describes the non-trivial asymptotic behavior of this spectral accumulation.     

Metrically Universal Generic Structures in Free Amalgamation Classes

We prove that each ∀₁ free amalgamation class K over a finite relational language L admits a countable generic structure M isometrically embedding all countable structuresin K relative to a fixed metric. We expand L by infinitely many binary predicates expressingdistance, and prove that the resulting expansion of K has a model companion axiomatizedby the first‐order theory of M. The model companion is non‐finitely axiomatizable, evenover a strong form of the axiom scheme of infinity.     

Free-knot spline approximation of stochastic processes

We study optimal approximation of stochastic processes by polynomial splines with free knots. The number of free knots is either a priori fixed or may depend on the particular trajectory. For the s-fold integrated Wiener process as well as for scalar diffusion processes we determine the asymptotic behavior of the average L_p-distance to the splines spaces, as the (expected) number of free knots tends to infinity.     

The Mass Shell in the Semi-Relativistic Pauli–Fierz Model

We consider the semi-relativistic Pauli–Fierz model for a single free electron interacting with the quantized radiation field. Employing a variant of Pizzo’s iterative analytic perturbation theory we construct a sequence of ground state eigenprojections of infra-red cutoff, dressing transformed fiber Hamiltonians and prove its convergence, as the cutoff goes to zero. Its limit is the ground state eigenprojection of a certain renormalized fiber Hamiltonian. The ground state energy is an exactly twofold degenerate eigenvalue of the renormalized Hamiltonian, while it is not an eigenvalue of the original fiber Hamiltonian unless the total momentum is zero. These results hold true, for total momenta inside a ball about zero of arbitrary radius ${\mathfrak{p} > 0}$ , provided that the coupling constant is sufficiently small depending on ${\mathfrak{p}}$ and the ultra-violet cutoff. Along the way we prove twice continuous differentiability and strict convexity of the ground state energy as a function of the total momentum inside that ball.     

GLOBAL EXISTENCE, UNIFORM DECAY AND EXPONENTIAL GROWTH FOR A CLASS OF SEMI-LINEAR WAVE EQUATION WITH STRONG DAMPING

In this paper, we consider the nonlinearly damped semi-linear wave equation associated with initial and Dirichlet boundary conditions. We prove the existence of a local weak solution and introduce a family of potential wells and discuss the invariants and vacuum isolating behavior of solutions. Furthermore, we prove the global existence of solutions in both cases which are polynomial and exponential decay in the energy space respectively, and the asymptotic behavior of solutions for the cases of potential well family with 0相似文献   

Translation-Invariant Bilinear Operators with Positive Kernels

We study L ^r (or L ^{r, ∞}) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on \mathbb Rⁿ{\mathbb {R}^n}. We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on \mathbb R²ⁿ{\mathbb R^{2n}}, while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some sufficient conditions on nonnegative kernels yielding boundedness for the corresponding operators on products of Lebesgue spaces. We also prove that, unlike the linear case where boundedness from L ¹ to L ¹ and from L ¹ to L ^{1, ∞} are equivalent properties, boundedness from L ¹ × L ¹ to L ^1/2 and from L ¹ × L ¹ to L ^{1/2, ∞} may not be equivalent properties for bilinear translation-invariant operators with nonnegative kernels.     

The Palais-Smale condition on contact type energy levels for convex Lagrangian systems

We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Mañé's critical value of the lift of the Lagrangian to the universal cover, c _u (L), almost all energy levels have conjugate points. We in addition prove that if an energy level is of contact type, projects onto M and $M\ne{\mathbb T}^2We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Ma?é's critical value of the lift of the Lagrangian to the universal cover, c _u (L), almost all energy levels have conjugate points.We in addition prove that if an energy level is of contact type, projects onto M and , then the free time action functional of L+k satisfies the Palais-Smale condition.Partially supported by Conacyt, Mexico, grant 36496-E.     

Geometric methods for nonlinear many-body quantum systems

Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body Schrödinger operators. In this paper we provide a formalism which also allows to study nonlinear systems. We start by defining a weak topology on many-body states, which appropriately describes the physical behavior of the system in the case of lack of compactness, that is when some particles are lost at infinity. We provide several important properties of this topology and use them to write a simple proof of the famous HVZ theorem in the repulsive case. In the second step we recall the method of geometric localization in Fock space as proposed by Dereziński and Gérard, and we relate this tool to our weak topology. We then provide several applications. We start by studying the so-called finite-rank approximation which consists in imposing that the many-body wavefunction can be expanded using finitely many one-body functions. We thereby emphasize geometric properties of Hartree-Fock states and prove nonlinear versions of the HVZ theorem, in the spirit of works of Friesecke. In the last section we study translation-invariant many-body systems comprising a nonlinear term, which effectively describes the interactions with a second system. As an example, we prove the existence of the multi-polaron in the Pekar-Tomasevich approximation, for certain values of the coupling constant.     

Three-manifolds with small L 2-norm of traceless-Ricci curvature pinching

In this paper, we consider a fourth-order gradient flow of the quadratic Riemannian functional ɛ of traceless Ricci curvature on closed 3 -manifolds with a fixed conformal class. We show that the L ²-curvature pinching locally conformally flat 3-manifolds can be deformed to space forms through such gradient flow. More precisely, for the suitable small initial energy functional ɛ, the gradient flow exists for all times and converges smoothly to space forms as the time goes to infinity. As a consequence, we prove the stability for any background metric whose such gradient flow converges to an Einstein metric.Mathematics Subject Classifications (2000): Primary: 53C21; Secondary: 58JOS.Communicated by: Claude LeBrun (Stony Brook)